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MAE 5130: VISCOUS FLOWS
Lecture 3: Kinematic Properties
August 24, 2010
Mechanical and Aerospace Engineering Department
Florida Institute of Technology
D. R. Kirk
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CHAPTER 1: CRITICAL READING
• 1-2 (all)
– Know how to derive Eq. (1-3)
• 1-3 (1-3.1 – 1-3.6, 1-3.8, 1-3.12 – 1-3.17)
– Understanding between Lagrangian and Eulerian viewpoints
– Detailed understanding of Figure 1-14
– Eq. (1-12) use of tan-1 vs. sin-1
– Familiarity with tensors
• 1-4 (all)
– Fluid boundary conditions: physical and mathematical understanding
• Comments
– Note error in Figure 1-14
– Problem 1-8 should read, ‘Using Eq. (1-3) for inviscid flow past a cylinder…’
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KINEMATIC PROPERTIES: TWO ‘VIEWS’ OF MOTION
1. Lagrangian Description
– Follow individual particle trajectories
– Choice in solid mechanics
– Control mass analyses
– Mass, momentum, and energy usually formulated for particles or systems of fixed identity (ex., F=d/dt(mV) is Lagrandian in nature)
2. Eulerian Description
– Study field as a function of position and time; not follow any specific particle paths
– Usually choice in fluid mechanics
– Control volume analyses
– Eulerian velocity vector field:
– Knowing scalars u, v, w as f(x,y,z,t) is a solution
ktzyxwjtzyxvitzyxutzyxVtrV ˆ,,,ˆ,,,ˆ,,,,,,,
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KINEMATIC PROPERTIES
• Let Q represent any property of the fluid (, T, p, etc.)
• Total differential change in Q
• Spatial increments
• Time derivative of Q of a particular elemental particle
• Substantial derivative, particle derivative or material derivative
• Particle acceleration vector– 9 spatial derivatives– 3 local (temporal) derivates
VVt
V
Dt
VD
QVt
Q
Dt
DQ
z
Qw
y
Qv
x
Qu
t
Q
dt
dQ
wdtdz
vdtdy
udtdx
dtt
Qdz
z
Qdy
y
Qdx
x
QdQ
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4 TYPES OF MOTION
• In fluid mechanics we are interested in general motion, deformation, and rate of deformation of particles
• Fluid element can undergo 4 types of motion or deformation:
1. Translation
2. Rotation
3. Shear strain
4. Extensional strain or dilatation
• We will show that all kinematic properties of fluid flow
– Acceleration
– Translation
– Angular velocity
– Rate of dilatation
– Shear strain
are directly related to fluid velocity vector V = (u, v, w)
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1. TRANSLATION
dx
dy
A
B C
D
y
x
+
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1. TRANSLATION
dx
dy
A
B C
D
A’
B’ C’
D’
udt
vdt
y
x
+
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2. ROTATION
dx
dy
A
B C
D
y
x
+
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2. ROTATION
• Angular rotation of element about z-axis is defined as the average counterclockwise rotation of the two sides BC and BA
– Or the rotation of the diagonal DB to B’D’
dx
dy
A
B C
D A’
B’
C’
D’
y
x
+
d
d
ddd z 2
1
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2. ROTATION
dydty
u
y
u
x
v
dt
d
dtx
v
dx
dxdtxv
d
dty
u
dy
dydtyu
d
ddd
z
z
2
1
tan
tan
2
1
1
1
A’
B’
C’
D’
d
d
y
x
+
dxdtx
v
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3. SHEAR STRAIN
dx
dy
A
B C
D
y
x
+
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3. SHEAR STRAIN
dx
dy
A
B C
D
y
x
+
d
d
• Defined as the average decrease of the angle between two lines which are initially perpendicular in the unstrained state (AB and BC)
dt
d
dt
d
dd
xy
2
12
1Shear-strain increment
Shear-strain rate
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COMMENTS: STRAIN VS. STRAIN RATE
• Strain is non-dimensional
– Example: Change in length L divided by initial length, L: L/L
– In solid mechanics this is often given the symbol , non-dimensional
– Recall Hooke’s Law: = E• Modulus of elasticity
• In fluid mechanics, we are interested in rates
– Example: Change in length L divided by initial length, L, per unit time: L/Lt gives units of [1/s]
– In fluid mechanics we will use the symbol for strain rate, [1/s]
– Strain rates will be written as velocity derivates
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4. EXTENSIONAL STRAIN (DILATATION)
dx
dy
A
B C
D
y
x
+
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4. EXTENSIONAL STRAIN (DILATATION)
dx
dy
A
B C
D
A’
B’ C’
D’• Extensional strain in x-direction is defined as the fractional increase in length of the
horizontal side of the element
y
x
+
dtx
u
dx
dxdxdtxu
dxdtxx
dxdtx
udx
Extensional strain in x-direction
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FIGURE 1-14: DISTORTION OF A MOVING FLUID ELEMENT
dxdtx
v
Not
e: M
ista
ke in
text
boo
k F
igur
e 1-
14
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COMMENTS ON ANGULAR ROTATION
• Recall: angular rotation of element about z-axis is defined as average counterclockwise rotation of two sides BC and BA
• BC has rotated CCW d• BA has rotated CW (-d)
• Overall CCW rotation since d > d• d and d both related to velocity derivates
through calculus limits
• Rates of angular rotation (angular velocity)
• 3 components of angular velocity vector ddt
• Very closely related to vorticity
• Recall: the vorticity, , is equal to twice the local angular velocity, d/dt (see example in Lecture 2)
dt
d
x
w
z
ud
z
v
y
wd
y
u
x
vd
dty
u
dydtyv
dx
dydtyu
d
dtx
v
dxdtxu
dx
dxdtxv
d
ddd
y
x
z
dt
dt
z
2
2
1
2
1
2
1
tanlim
tanlim
2
1
1
0
1
0
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COMMENTS ON SHEAR STRAIN
• Recall: defined as the average decrease of the angle between two lines which are initially perpendicular in the unstrained state (AB and BC)
• Shear-strain rates
• Shear-strain rates are symmetric
jiij
zx
yz
xy
dx
dw
dz
du
dz
dv
dy
dw
dy
du
dx
dv
dt
d
dt
d
dd
2
1
2
1
2
1
2
1
2
1
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COMMENTS ON EXTENSIONAL STRAIN RATES
• Recall: the extensional strain in the x-direction is defined as the fractional increase in length of the horizontal side of the element
• Extensional strains
z
w
y
vx
u
dtx
u
dx
dxdxdtxu
dxdt
zz
yy
xx
xx
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STRAIN RATE TENSOR
• Taken together, shear and extensional strain rates constitute a symmetric 2nd order tensor
• Tensor components vary with change of axes x, y, z
• Follows transformation laws of symmetric tensors
• For all symmetric tensors there exists one and only one set of axes for which the off-diagonal terms (the shear-strain rates) vanish
– These are called the principal axes
3
2
1
00
00
00
zzzyzx
yzyyyx
xzxyxx
ij
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USEFUL SHORT-HAND NOTATION
• Short-hand notation
– i and j are any two coordinate directions
• Vector can be split into two parts
– Symmetric
– Antisymmetric
• Each velocity derivative can be resolved into a strain rate () plus an angular velocity (d/dt)
dt
d
x
u
uuuuu
x
uu
ijij
j
i
ijjiijjiji
j
iji
,,,,,
,
2
1
2
1
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DEVELOPMENT OF N/S EQUATIONS: ACCELERATION
surface
body
externalsurfacebody
externalsurfacebody
fgDt
VD
gf
fffDt
VD
Dt
VDa
ffffV
Fa
Fam
• Momentum equation, Newton
• Concerned with:
– Body forces
• Gravity
• Electromagnetic potential
– Surface forces
• Friction (shear, drag)
• Pressure
– External forces
• Eulerian description of acceleration
• Substitution in to momentum
• Recall that body forces apply to entire mass of fluid element
• Now ready to develop detailed expressions for surface forces (and how they related to strain, which are related to velocity derivatives)
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SUMMARY
• All kinematic properties of fluid flow
– Acceleration: DV/Dt
– Translation: udt, vdt, wdt
– Angular velocity: d/dt
• dx/dt, dy/dt, dz/dt
• Also related to vorticity
– Shear-strain rate: xy=yx, xz=zx, yz=zy
– Rate of dilatation: xx, yy, zz
are directly related to the fluid velocity vector V = (u, v, w)
• Translation and angular velocity do not distort the fluid element
• Strains (shear and dilation) distort the fluid element and cause viscous stresses
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